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What is the relationship between DTFT and continuous fourier transform?

As title says, what is the relationship between DTFT and continuous fourier transform? Let's say there is continious signal $f(t)$. Continuous Fourier transform convert this into $F(\omega)$. Now let ...
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28 views

Help understanding Wiener filtering formula

I would like some help interpreting the following formula, equation 1 from this paper: https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/strela.pdf $\hat{X} = ...
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11 views

How to work with 4-1 multiplexer In digital logic?

Here is my image of multiplexer, http://d18khu5s3lkxd9.cloudfront.net//wp-content/uploads/2014/04/GATECS2014Q55.png and this one ...
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23 views

Understanding, Non-Negative Sparse Coding algorithm

I have a question regarding sparse coding, Non-negative sparse coding. Iterate until convergence: $ \mathbf{A_i} \leftarrow \arg \! \min_{A \geq 0} || \mathbf{X}_i - \mathbf{B}_i\mathbf{A}||_F^2 + ...
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62 views

Fourier transform and splitting frequency range into 4 channels

I have code example that divides audio frequency into 6 channels. It uses Fast Fourier Transform (FFT). Algorithm process the frequency range using 6 capture[x] samples based on the range of n between ...
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79 views

Discrete Fourier Transform question

Let $R_{M\times N}$ be a space of size $M\times N$. Define the 2D Discrete Fourier Transform of $f\in R_{M\times N}$ to be \begin{equation} ...
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133 views

About integrating product of two sinc function using Fourier transform

So the problem is which I think is pretty straight-foward by using Fourier transform and convolution property of two sinc functions and evaluating the convolution at 5. However, I got sinc(t) for ...
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43 views

Deriving the autocorrelation function for the ARMA model

Definitions The ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. Question To derive the ...
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45 views

Yule walker equation limited matrix size

Definitions For an ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. It is straightforward to show that ...
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30 views

If the signal's frequency is multiples of the first harmonic frequency, transform method similar to DFT but use less number of samples?

Suppose that a continuous signal $f(t)$ has the first harmonic frequency $f_1$. $f(t)$'s frequencies that are not integer multiples of $f_1$ are known to have zero signal magnitude $|F(\omega)|$. This ...
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33 views

DSP Time domain and frequency domain

I'm new here and wish to say hello to this great community. I'm starting to learn DSP, I don't have a lot of Maths background but I'm trying to learn. I am new to DSP too and I am reading this great ...
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29 views

Which of arithmetic, geometric or harmonic mean is the most appropriate in this case?

I have a software that periodically detects tempo out of an audio signal and I would like to compute the average tempo out of all the generated values. Example: ...
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87 views

$E[x_i^2 x_j^2]$ for white Gaussian noise

If $x_n$ is a discrete time random signal and is white Gaussian noise (ergodic and WSS) so $$E[x_n x_{n+l}]=\sigma ^2 \delta (l)$$ and $$E[x_n]=0$$ Where $n \in \mathbb{R}$ and $l\in\mathbb{R}$ ...
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100 views

matlab problem - removing frequencies after FFT, signal processing

I want to stress that this is not a coding problem, my problem is that i don't fully understand the mathematics surrounding the subject and that's why I believe I have a problem. I was given an ...
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50 views

convolution and associativity

Ok Let talk about this,... I am now so confused. 1-$$\mathcal{F}\Big\{c(x-x_0)b(x-x_0)\Big\}=\mathcal{F}\Big\{c(x-x_0)\Big\}\circ\mathcal{F}\Big\{b(x-x_0)\Big\}\\=\Bigg[e^{-2ix_0y}C(y) ...
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92 views

How one can show $P(ax+n|x)=P(n)$? [closed]

Let $x$ be a signal and $n$ be an independent noise. How one can show $P(ax+n|x)=P(n)$? Thanks. Well, let $y=ax+n$, so we have $n=y-ax$. Now if the probability density function (PDF) of $n$ for ...
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62 views

Convolution sum. Compute $y[n]=x[n]\ast h[n]$

Compute $y[n]=x[n]\ast h[n]$ $x[n]=(-\frac{1}{2})^2u[n-4]$ $h[n]=4^nu[2-n]$ In this question, when I try to calculate the convolution sum. I face with: ...
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29 views

Calculate Distance between Fourier Transforms

I'm working with signal data (specifically data from accelerators and gyroscopes), and I take their Fourier transforms to get a better idea of the dominant frequencies. I'd like to compare the ...
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24 views

DFT by $n$ samples of a continuous periodic signal with more than $n$ frequencies

It is known that if we only have $n$ samples and take DFT, we only get at most $n$ distinct frequency data. But let's say that there is a continuous periodic signal with more than $n$ frequencies, ...
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36 views

If a signal is periodic, can the error of approximation by Discrete Fourier Transform be avoided when using finite number of samples?

As title says, if a signal $f(t)$ is periodic, can approximation errors of approximation by discrete Fourier transform (DFT) be avoided when only finite number of samples are used?
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34 views

Why are discrete-time Fourier series and discrete Fourier transform only defined on integer $k$?

In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$. But why is it the case that ...
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37 views

Using Discrete Fourier trasform of the samples of a continuous/periodic signal to obtain frequency data similar to FT of the original signal

Suppose we have a continuous and periodic real-valued 1D signal $f(t)$. Let us say we obtain finite number of samples $f(n)$ from $f(t)$. Is there a way to take discrete Fourier transform of $f(n)$ ...
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42 views

How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
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27 views

Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
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25 views

How to find Bilateral Laplace Transform of $e^{at}$ Using Changing of the Time Horizon

Ok, this has me a bit stumped. In my class the teacher "showed" us how to find the bilateral Laplace transform of x(t)=$e^{at}$ where $-\infty<t<\infty$. Breaking them into the two parts ...
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23 views

Can we obtain Fourier transform of a continuous signal using finite number of samples of the signal with known frequency cutoff?

Suppose that there is a continuous signal with highest frequency known. Is there a way so that we only sample the signal finite times and obtain the Fourier transform of the original signal (which ...
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24 views

Frequency spectrum of signal and is it real?

$x(t) = 2 + 5 cos(-t + \pi/4) - 2sin(3t + 5) + 3(cos(5 t + \pi/2).cos(4t) - e^je^t $ a) To find Fourier series coefficients of the following signal I need to use inverse Euler formula. But I need ...
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43 views

What is the connection between random variables and time series?

I always felt that there was a disconnect between random variable and time series. Clearly, random variable and time series can both be treated with statistical methods. First order, second order ...
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32 views

Two forms of cross-correlation

Wikipedia and MATLAB defines cross-correlation in this way. In time series analysis (P21), it defines cross-correlation upon cross-covariance: Let $\{X_t\}$ and $\{Y_t\}$ be two time series, ...
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27 views

Algorithm - return aliasing frequency

I posted this question on StackOverflow as well (link), but it is somewhere between a math question and a programming question (I'm looking for some formulas regarding aliasing frequencies and I want ...
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64 views

Fourier transform: noise and variance

I wrote a short program to generate $N$ samples of a sinusoid with some noise (ie: $$ f(t) = \cos(2\pi t) + 0.1 * \text{noise}(t) $$ where $\text{noise}(t)$ is chosen uniformly from $[-1 , 1]$. ...
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68 views

Orthonormal basis from Riesz basis

This question is with respect to Theorem 7.1 of Mallat's Wavelet Tour text. It is a follow-up of sorts to a previous question. Preliminaries Suppose I have a set $\{\theta(t-n)\}_{n \in \mathbb{Z}}$ ...
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67 views

Hessian Of Convolution's Quadratic Form

For the discrete inputs $\mathbf{x} \in \mathbb{C}^{M}$ and $\mathbf{y} \in \mathbb{C}^{N}$, I want to find the Hessian of $\Vert x \ast y \Vert_2^2$, where $\ast$ is the discrete convolution ...
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43 views

Discrete-time sinusoids with same frequency

I've read that sine waves of the form $x_n = \sin(w_{0}n)$, with frequencies $w_{0}$ and $w_{0} + 2\pi$, are indistinguishable from each other when considering discrete time. The book gives ...
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65 views

How to plot phasors of signals?

I have 3 singals and I'm trying to plot their phasors and their sum. I need to plot them end to end to demonstrate phasor addition. That is, the first phasor must start from the origin. The second ...
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22 views

A filter with frequency response $H(f)=\operatorname{sinc}(f).$

In signal processing, a sinc filter is an idealized filter that have the following frequency response $$H(f) = \mathrm{rect} \left( \frac{f}{2B} \right)$$ that is the rectangular function. In the real ...
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20 views

Alternative Function Definitions for the Square Wave signal

Are there any other function definitions for the Square Wave signal rather than the : and those referred to Wikipedia ?
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21 views

Time invariace of a linear system dependent on a particular time instant

$$y[n]=x[n]+35*x[n-1]+x[0]$$ Is this system time invariant? I am under the impression that $x[0]$ can be considered a constant. Am I right?
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21 views

how to find out how many Fourier coefficients there are (which are not zeros)

given a real periodic (with period $T_0$) signal $x(t)$ with fourier transform in which $$X(jw)=0\ \ \forall |w|\ge {6\pi \over T_0}$$ I know that the fourier series will have finite coefficients (5 ...
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48 views

Fourier transform of a certain equality / discrete time Fourier transform of Dirac delta?

This comes from Stephane Mallat's Wavelet Tour text; however, I will phrase my question independently of it. I apologize that this is sort of long-winded. We have a function $f$ which satisfies the ...
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20 views

How to estimate parameters of a parametric function if its values for a set of arguments are known?

Suppose we have a parametric function $F(\alpha_1, ..., \alpha_{N_p}, \mathbf{x})$. For a set of arguments $\mathbf{x_1}$ ... $\mathbf{x_N}$ it's values $F(\mathbf{x_1})$ .... $F(\mathbf{x_N})$ are ...
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Fourier transform of integral function

A function $s(t)$ is defined by $s(t)=\int_x p(t-cx)dx$ where $\tau = cx$ is a time variable and $t\neq \tau$. What is the Fourier transform, $S(\omega)$, of the function $s(t)$? I know that for a ...
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Is “quantum” a correct term for the subsets used by a quantization function?

A quantizer is a many-to-few map. Its domain then is sets. I've heard those sets referred to as quanta (the plural of quantum). That usage seems to agree with what I understand to be the ...
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What is the term for the value from the set of values in a quantum that is closest to the aliased value of the quantum?

Signal quantization results in aliasing of the quanta. Is there a term for the value in the quantized set that is closest to the aliased value of the quantum? Something like "nearest neighbor" or ...
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Regarding the unilateral Laplace transform of LTI systems

Consider an LTI system described by the following differential equation, $$ \sum_{k=0}^{N}a_k\frac{d}{dt^k}y(t) = \sum_{k=0}^{M}b_k\frac{d}{dt^k}x(t) $$ With initial conditions, $$ y(t)|_{t=0}, ...
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187 views

compare lines and recognize similar ones

how can I find similar patterns in a line if I got a "template-line"? In this example, if I got the template (red), how can I find out that there are two occurences in the green one? The lines ...
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1answer
55 views

Convolution Properties

I have a quick question about certain algebraic properties of convolution. If I have 3 functions $f(x)$, $g(x)$ and $h(x)$, is the following true? $\Big[ f(x) . g(x)\Big] \circ h(x) = \Big[f(x) \circ ...
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60 views

Frequency response of a linear, shift-variant system

I am working my way through recorded lectures and a textbook related to DSP, and have come across a question that I am not sure how to answer. This is probably just due to how new I am to these ...
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1answer
29 views

Determine a time signal from another time signal

The given time signal is: $$u(t) = -3\sigma(t+4) + 6\sigma(t) - 3\sigma(t-4)$$ $\sigma$ - unit step function The same signal can be describes with the following mathematical relation between ...
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92 views

Fourier spectrum reflected across origin and Nyquist frequency

Recently I've been trying to figure out what's the point of negative frequencies produced by the fourier transform. One answer was it's just there to make calculations more elegant. It could be ...